Smoothness of stabilisers in generic characteristic

Let R be a commutative unital ring. Given a finitely presented affine R-group scheme G acting on a finitely presented separated scheme X over R, we show that there is a prime p0 such that for any R-algebra k which is a field of characteristic p p0, the centraliser in Gk of any closed subscheme of Xk is smooth. When X is not necessarily separated we show similarly that for any closed finitely presented subscheme Y X there is a p1 depending on Y such that when k has characteristic p p1, the normaliser of Yk in Gk is smooth. For the proof, we may assume k is algebraically closed, whence we prove these results using the Lefschetz principle together with careful application of Gröbner basis techniques, and using a suitable notion of the complexity of an action. We apply our results to demonstrate that the Kostant-Kirillov-Souriau theorem holds for Lie algebras of algebraic groups in large positive characteristics: the coadjoint module of every such Lie algebra decomposes as a disjoint union of symplectic varieties, each of which is a coadjoint orbit.